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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.autodiff"></a><a class="link" href="autodiff.html" title="Automatic Differentiation">Automatic Differentiation</a>
</h2></div></div></div>
<h2>
<a name="math_toolkit.autodiff.h0"></a>
      <span class="phrase"><a name="math_toolkit.autodiff.synopsis"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.synopsis">Synopsis</a>
    </h2>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">differentiation</span><span class="special">/</span><span class="identifier">autodiff</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>

<span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span>
<span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>
<span class="keyword">namespace</span> <span class="identifier">differentiation</span> <span class="special">{</span>

<span class="comment">// Function returning a single variable of differentiation. Recommended: Use auto for type.</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">Order</span><span class="special">,</span> <span class="identifier">size_t</span><span class="special">...</span> <span class="identifier">Orders</span><span class="special">&gt;</span>
<span class="identifier">autodiff_fvar</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">,</span> <span class="identifier">Orders</span><span class="special">...&gt;</span> <span class="identifier">make_fvar</span><span class="special">(</span><span class="identifier">RealType</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">ca</span><span class="special">);</span>

<span class="comment">// Function returning multiple independent variables of differentiation in a std::tuple.</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">size_t</span><span class="special">...</span> <span class="identifier">Orders</span><span class="special">,</span> <span class="keyword">typename</span><span class="special">...</span> <span class="identifier">RealTypes</span><span class="special">&gt;</span>
<span class="keyword">auto</span> <span class="identifier">make_ftuple</span><span class="special">(</span><span class="identifier">RealTypes</span> <span class="keyword">const</span><span class="special">&amp;...</span> <span class="identifier">ca</span><span class="special">);</span>

<span class="comment">// Type of combined autodiff types. Recommended: Use auto for return type (C++14).</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">typename</span><span class="special">...</span> <span class="identifier">RealTypes</span><span class="special">&gt;</span>
<span class="keyword">using</span> <span class="identifier">promote</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">promote_args_n</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">RealTypes</span><span class="special">...&gt;::</span><span class="identifier">type</span><span class="special">;</span>

<span class="keyword">namespace</span> <span class="identifier">detail</span> <span class="special">{</span>

<span class="comment">// Single autodiff variable. Use make_fvar() or make_ftuple() to instantiate.</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">Order</span><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">fvar</span> <span class="special">{</span>
 <span class="keyword">public</span><span class="special">:</span>
  <span class="comment">// Query return value of function to get the derivatives.</span>
  <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span><span class="special">...</span> <span class="identifier">Orders</span><span class="special">&gt;</span>
  <span class="identifier">get_type_at</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">sizeof</span><span class="special">...(</span><span class="identifier">Orders</span><span class="special">)</span> <span class="special">-</span> <span class="number">1</span><span class="special">&gt;</span> <span class="identifier">derivative</span><span class="special">(</span><span class="identifier">Orders</span><span class="special">...</span> <span class="identifier">orders</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>

  <span class="comment">// All of the arithmetic and comparison operators are overloaded.</span>
  <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType2</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">Order2</span><span class="special">&gt;</span>
  <span class="identifier">fvar</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">+=(</span><span class="identifier">fvar</span><span class="special">&lt;</span><span class="identifier">RealType2</span><span class="special">,</span> <span class="identifier">Order2</span><span class="special">&gt;</span> <span class="keyword">const</span><span class="special">&amp;);</span>

  <span class="identifier">fvar</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">+=(</span><span class="identifier">root_type</span> <span class="keyword">const</span><span class="special">&amp;);</span>

  <span class="comment">// ...</span>
<span class="special">};</span>

<span class="comment">// Standard math functions are overloaded and called via argument-dependent lookup (ADL).</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">Order</span><span class="special">&gt;</span>
<span class="identifier">fvar</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">&gt;</span> <span class="identifier">floor</span><span class="special">(</span><span class="identifier">fvar</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">&gt;</span> <span class="keyword">const</span><span class="special">&amp;);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">Order</span><span class="special">&gt;</span>
<span class="identifier">fvar</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">&gt;</span> <span class="identifier">exp</span><span class="special">(</span><span class="identifier">fvar</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">&gt;</span> <span class="keyword">const</span><span class="special">&amp;);</span>

<span class="comment">// ...</span>

<span class="special">}</span>  <span class="comment">// namespace detail</span>

<span class="special">}</span>  <span class="comment">// namespace differentiation</span>
<span class="special">}</span>  <span class="comment">// namespace math</span>
<span class="special">}</span>  <span class="comment">// namespace boost</span>
</pre>
<h2>
<a name="math_toolkit.autodiff.h1"></a>
      <span class="phrase"><a name="math_toolkit.autodiff.description"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.description">Description</a>
    </h2>
<p>
      Autodiff is a header-only C++ library that facilitates the <a href="https://en.wikipedia.org/wiki/Automatic_differentiation" target="_top">automatic
      differentiation</a> (forward mode) of mathematical functions of single
      and multiple variables.
    </p>
<p>
      This implementation is based upon the <a href="https://en.wikipedia.org/wiki/Taylor_series" target="_top">Taylor
      series</a> expansion of an analytic function <span class="emphasis"><em>f</em></span> at
      the point <span class="emphasis"><em>x<sub>0</sub></em></span>:
    </p>
<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../equations/autodiff/taylor_series.svg"></span>
        </p></blockquote></div></blockquote></div>
<p>
      The essential idea of autodiff is the substitution of numbers with polynomials
      in the evaluation of <span class="emphasis"><em>f(x<sub>0</sub>)</em></span>. By substituting the number
      <span class="emphasis"><em>x<sub>0</sub></em></span> with the first-order polynomial <span class="emphasis"><em>x<sub>0</sub>+ε</em></span>,
      and using the same algorithm to compute <span class="emphasis"><em>f(x<sub>0</sub>+ε)</em></span>,
      the resulting polynomial in <span class="emphasis"><em>ε</em></span> contains the function's
      derivatives <span class="emphasis"><em>f'(x<sub>0</sub>)</em></span>, <span class="emphasis"><em>f''(x<sub>0</sub>)</em></span>, <span class="emphasis"><em>f'''(x<sub>0</sub>)</em></span>,
      ... within the coefficients. Each coefficient is equal to the derivative of
      its respective order, divided by the factorial of the order.
    </p>
<p>
      In greater detail, assume one is interested in calculating the first <span class="emphasis"><em>N</em></span>
      derivatives of <span class="emphasis"><em>f</em></span> at <span class="emphasis"><em>x<sub>0</sub></em></span>. Without loss
      of precision to the calculation of the derivatives, all terms <span class="emphasis"><em>O(ε<sup>N+1</sup>)</em></span>
      that include powers of <span class="emphasis"><em>ε</em></span> greater than <span class="emphasis"><em>N</em></span>
      can be discarded. (This is due to the fact that each term in a polynomial depends
      only upon equal and lower-order terms under arithmetic operations.) Under these
      truncation rules, <span class="emphasis"><em>f</em></span> provides a polynomial-to-polynomial
      transformation:
    </p>
<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../equations/autodiff/polynomial_transform.svg"></span>
        </p></blockquote></div></blockquote></div>
<p>
      C++'s ability to overload operators and functions allows for the creation of
      a class <code class="computeroutput"><span class="identifier">fvar</span></code> (<span class="underline">f</span>orward-mode
      autodiff <span class="underline">var</span>iable) that represents polynomials
      in <span class="emphasis"><em>ε</em></span>. Thus the same algorithm <span class="emphasis"><em>f</em></span>
      that calculates the numeric value of <span class="emphasis"><em>y<sub>0</sub>=f(x<sub>0</sub>)</em></span>, when written
      to accept and return variables of a generic (template) type, is also used to
      calculate the polynomial <span class="emphasis"><em>Σ<sub>n</sub>y<sub>n</sub>ε<sup>n</sup>=f(x<sub>0</sub>+ε)</em></span>.
      The derivatives <span class="emphasis"><em>f<sup>(n)</sup>(x<sub>0</sub>)</em></span> are then found from the product
      of the respective factorial <span class="emphasis"><em>n!</em></span> and coefficient <span class="emphasis"><em>y<sub>n</sub></em></span>:
    </p>
<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../equations/autodiff/derivative_formula.svg"></span>
        </p></blockquote></div></blockquote></div>
<h2>
<a name="math_toolkit.autodiff.h2"></a>
      <span class="phrase"><a name="math_toolkit.autodiff.examples"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.examples">Examples</a>
    </h2>
<h3>
<a name="math_toolkit.autodiff.h3"></a>
      <span class="phrase"><a name="math_toolkit.autodiff.example-single-variable"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.example-single-variable">Example
      1: Single-variable derivatives</a>
    </h3>
<h4>
<a name="math_toolkit.autodiff.h4"></a>
      <span class="phrase"><a name="math_toolkit.autodiff.calculate_derivatives_of_f_x_x_s"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.calculate_derivatives_of_f_x_x_s">Calculate
      derivatives of <span class="emphasis"><em>f(x)=x<sup>4</sup></em></span> at <span class="emphasis"><em>x</em></span>=2.</a>
    </h4>
<p>
      In this example, <code class="computeroutput"><span class="identifier">make_fvar</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span>
      <span class="identifier">Order</span><span class="special">&gt;(</span><span class="number">2.0</span><span class="special">)</span></code> instantiates
      the polynomial 2+<span class="emphasis"><em>ε</em></span>. The <code class="computeroutput"><span class="identifier">Order</span><span class="special">=</span><span class="number">5</span></code> means that
      enough space is allocated (on the stack) to hold a polynomial of up to degree
      5 during the proceeding computation.
    </p>
<p>
      Internally, this is modeled by a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span><span class="number">6</span><span class="special">&gt;</span></code> whose elements <code class="computeroutput"><span class="special">{</span><span class="number">2</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span>
      <span class="number">0</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span> <span class="number">0</span><span class="special">}</span></code>
      correspond to the 6 coefficients of the polynomial upon initialization. Its
      fourth power, at the end of the computation, is a polynomial with coefficients
      <code class="computeroutput"><span class="identifier">y</span> <span class="special">=</span>
      <span class="special">{</span><span class="number">16</span><span class="special">,</span>
      <span class="number">32</span><span class="special">,</span> <span class="number">24</span><span class="special">,</span> <span class="number">8</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span>
      <span class="number">0</span><span class="special">}</span></code>. The
      derivatives are obtained using the formula <span class="emphasis"><em>f<sup>(n)</sup>(2)=n!*y[n]</em></span>.
    </p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">differentiation</span><span class="special">/</span><span class="identifier">autodiff</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span>
<span class="identifier">T</span> <span class="identifier">fourth_power</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">{</span>
  <span class="identifier">T</span> <span class="identifier">x4</span> <span class="special">=</span> <span class="identifier">x</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">;</span>  <span class="comment">// retval in operator*() uses x4's memory via NRVO.</span>
  <span class="identifier">x4</span> <span class="special">*=</span> <span class="identifier">x4</span><span class="special">;</span>      <span class="comment">// No copies of x4 are made within operator*=() even when squaring.</span>
  <span class="keyword">return</span> <span class="identifier">x4</span><span class="special">;</span>     <span class="comment">// x4 uses y's memory in main() via NRVO.</span>
<span class="special">}</span>

<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> <span class="special">{</span>
  <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">differentiation</span><span class="special">;</span>

  <span class="keyword">constexpr</span> <span class="keyword">unsigned</span> <span class="identifier">Order</span> <span class="special">=</span> <span class="number">5</span><span class="special">;</span>                  <span class="comment">// Highest order derivative to be calculated.</span>
  <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">make_fvar</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">&gt;(</span><span class="number">2.0</span><span class="special">);</span>  <span class="comment">// Find derivatives at x=2.</span>
  <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">fourth_power</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
  <span class="keyword">for</span> <span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;=</span> <span class="identifier">Order</span><span class="special">;</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
    <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"y.derivative("</span> <span class="special">&lt;&lt;</span> <span class="identifier">i</span> <span class="special">&lt;&lt;</span> <span class="string">") = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">y</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="identifier">i</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
  <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
<span class="special">}</span>
<span class="comment">/*
Output:
y.derivative(0) = 16
y.derivative(1) = 32
y.derivative(2) = 48
y.derivative(3) = 48
y.derivative(4) = 24
y.derivative(5) = 0
*/</span>
</pre>
<p>
      The above calculates
    </p>
<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../equations/autodiff/example1.svg"></span>
        </p></blockquote></div></blockquote></div>
<h3>
<a name="math_toolkit.autodiff.h5"></a>
      <span class="phrase"><a name="math_toolkit.autodiff.example-multiprecision"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.example-multiprecision">Example
      2: Multi-variable mixed partial derivatives with multi-precision data type</a>
    </h3>
<h4>
<a name="math_toolkit.autodiff.h6"></a>
      <span class="phrase"><a name="math_toolkit.autodiff.calculate_autodiff_equation_mixe"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.calculate_autodiff_equation_mixe">Calculate
      <span class="inlinemediaobject"><img src="../../equations/autodiff/mixed12.svg"></span> with a precision of about 50 decimal digits, where <span class="inlinemediaobject"><img src="../../equations/autodiff/example2f.svg"></span>.</a>
    </h4>
<p>
      In this example, <code class="computeroutput"><span class="identifier">make_ftuple</span><span class="special">&lt;</span><span class="identifier">float50</span><span class="special">,</span> <span class="identifier">Nw</span><span class="special">,</span>
      <span class="identifier">Nx</span><span class="special">,</span> <span class="identifier">Ny</span><span class="special">,</span> <span class="identifier">Nz</span><span class="special">&gt;(</span><span class="number">11</span><span class="special">,</span>
      <span class="number">12</span><span class="special">,</span> <span class="number">13</span><span class="special">,</span> <span class="number">14</span><span class="special">)</span></code> returns a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">tuple</span></code> of
      4 independent <code class="computeroutput"><span class="identifier">fvar</span></code> variables,
      with values of 11, 12, 13, and 14, for which the maximum order derivative to
      be calculated for each are 3, 2, 4, 3, respectively. The order of the variables
      is important, as it is the same order used when calling <code class="computeroutput"><span class="identifier">v</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="identifier">Nw</span><span class="special">,</span>
      <span class="identifier">Nx</span><span class="special">,</span> <span class="identifier">Ny</span><span class="special">,</span> <span class="identifier">Nz</span><span class="special">)</span></code> in the example below.
    </p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">differentiation</span><span class="special">/</span><span class="identifier">autodiff</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">cpp_bin_float</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>

<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">differentiation</span><span class="special">;</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">W</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">X</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Y</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Z</span><span class="special">&gt;</span>
<span class="identifier">promote</span><span class="special">&lt;</span><span class="identifier">W</span><span class="special">,</span> <span class="identifier">X</span><span class="special">,</span> <span class="identifier">Y</span><span class="special">,</span> <span class="identifier">Z</span><span class="special">&gt;</span> <span class="identifier">f</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">W</span><span class="special">&amp;</span> <span class="identifier">w</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">X</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Y</span><span class="special">&amp;</span> <span class="identifier">y</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Z</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">{</span>
  <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
  <span class="keyword">return</span> <span class="identifier">exp</span><span class="special">(</span><span class="identifier">w</span> <span class="special">*</span> <span class="identifier">sin</span><span class="special">(</span><span class="identifier">x</span> <span class="special">*</span> <span class="identifier">log</span><span class="special">(</span><span class="identifier">y</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">w</span> <span class="special">*</span> <span class="identifier">z</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">*</span> <span class="identifier">y</span><span class="special">)))</span> <span class="special">+</span> <span class="identifier">w</span> <span class="special">*</span> <span class="identifier">w</span> <span class="special">/</span> <span class="identifier">tan</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span>
<span class="special">}</span>

<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> <span class="special">{</span>
  <span class="keyword">using</span> <span class="identifier">float50</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_bin_float_50</span><span class="special">;</span>

  <span class="keyword">constexpr</span> <span class="keyword">unsigned</span> <span class="identifier">Nw</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>  <span class="comment">// Max order of derivative to calculate for w</span>
  <span class="keyword">constexpr</span> <span class="keyword">unsigned</span> <span class="identifier">Nx</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span>  <span class="comment">// Max order of derivative to calculate for x</span>
  <span class="keyword">constexpr</span> <span class="keyword">unsigned</span> <span class="identifier">Ny</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span>  <span class="comment">// Max order of derivative to calculate for y</span>
  <span class="keyword">constexpr</span> <span class="keyword">unsigned</span> <span class="identifier">Nz</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>  <span class="comment">// Max order of derivative to calculate for z</span>
  <span class="comment">// Declare 4 independent variables together into a std::tuple.</span>
  <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">variables</span> <span class="special">=</span> <span class="identifier">make_ftuple</span><span class="special">&lt;</span><span class="identifier">float50</span><span class="special">,</span> <span class="identifier">Nw</span><span class="special">,</span> <span class="identifier">Nx</span><span class="special">,</span> <span class="identifier">Ny</span><span class="special">,</span> <span class="identifier">Nz</span><span class="special">&gt;(</span><span class="number">11</span><span class="special">,</span> <span class="number">12</span><span class="special">,</span> <span class="number">13</span><span class="special">,</span> <span class="number">14</span><span class="special">);</span>
  <span class="keyword">auto</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">w</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">get</span><span class="special">&lt;</span><span class="number">0</span><span class="special">&gt;(</span><span class="identifier">variables</span><span class="special">);</span>  <span class="comment">// Up to Nw derivatives at w=11</span>
  <span class="keyword">auto</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">get</span><span class="special">&lt;</span><span class="number">1</span><span class="special">&gt;(</span><span class="identifier">variables</span><span class="special">);</span>  <span class="comment">// Up to Nx derivatives at x=12</span>
  <span class="keyword">auto</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">get</span><span class="special">&lt;</span><span class="number">2</span><span class="special">&gt;(</span><span class="identifier">variables</span><span class="special">);</span>  <span class="comment">// Up to Ny derivatives at y=13</span>
  <span class="keyword">auto</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">z</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">get</span><span class="special">&lt;</span><span class="number">3</span><span class="special">&gt;(</span><span class="identifier">variables</span><span class="special">);</span>  <span class="comment">// Up to Nz derivatives at z=14</span>
  <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">v</span> <span class="special">=</span> <span class="identifier">f</span><span class="special">(</span><span class="identifier">w</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">z</span><span class="special">);</span>
  <span class="comment">// Calculated from Mathematica symbolic differentiation.</span>
  <span class="identifier">float50</span> <span class="keyword">const</span> <span class="identifier">answer</span><span class="special">(</span><span class="string">"1976.319600747797717779881875290418720908121189218755"</span><span class="special">);</span>
  <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">float50</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">)</span>
            <span class="special">&lt;&lt;</span> <span class="string">"mathematica   : "</span> <span class="special">&lt;&lt;</span> <span class="identifier">answer</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span>
            <span class="special">&lt;&lt;</span> <span class="string">"autodiff      : "</span> <span class="special">&lt;&lt;</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="identifier">Nw</span><span class="special">,</span> <span class="identifier">Nx</span><span class="special">,</span> <span class="identifier">Ny</span><span class="special">,</span> <span class="identifier">Nz</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span>
            <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span>
            <span class="special">&lt;&lt;</span> <span class="string">"relative error: "</span> <span class="special">&lt;&lt;</span> <span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="identifier">Nw</span><span class="special">,</span> <span class="identifier">Nx</span><span class="special">,</span> <span class="identifier">Ny</span><span class="special">,</span> <span class="identifier">Nz</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">answer</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span><span class="special">;</span>
  <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
<span class="special">}</span>
<span class="comment">/*
Output:
mathematica   : 1976.3196007477977177798818752904187209081211892188
autodiff      : 1976.3196007477977177798818752904187209081211892188
relative error: 2.67e-50
*/</span>
</pre>
<h3>
<a name="math_toolkit.autodiff.h7"></a>
      <span class="phrase"><a name="math_toolkit.autodiff.example-black_scholes"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.example-black_scholes">Example
      3: Black-Scholes Option Pricing with Greeks Automatically Calculated</a>
    </h3>
<h4>
<a name="math_toolkit.autodiff.h8"></a>
      <span class="phrase"><a name="math_toolkit.autodiff.calculate_greeks_directly_from_t"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.calculate_greeks_directly_from_t">Calculate
      greeks directly from the Black-Scholes pricing function.</a>
    </h4>
<p>
      Below is the standard Black-Scholes pricing function written as a function
      template, where the price, volatility (sigma), time to expiration (tau) and
      interest rate are template parameters. This means that any greek based on these
      4 variables can be calculated using autodiff. The below example calculates
      delta and gamma where the variable of differentiation is only the price. For
      examples of more exotic greeks, see <code class="computeroutput"><span class="identifier">example</span><span class="special">/</span><span class="identifier">black_scholes</span><span class="special">.</span><span class="identifier">cpp</span></code>.
    </p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">differentiation</span><span class="special">/</span><span class="identifier">autodiff</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>

<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">;</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">differentiation</span><span class="special">;</span>

<span class="comment">// Equations and function/variable names are from</span>
<span class="comment">// https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks</span>

<span class="comment">// Standard normal cumulative distribution function</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">X</span><span class="special">&gt;</span>
<span class="identifier">X</span> <span class="identifier">Phi</span><span class="special">(</span><span class="identifier">X</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">{</span>
  <span class="keyword">return</span> <span class="number">0.5</span> <span class="special">*</span> <span class="identifier">erfc</span><span class="special">(-</span><span class="identifier">one_div_root_two</span><span class="special">&lt;</span><span class="identifier">X</span><span class="special">&gt;()</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">);</span>
<span class="special">}</span>

<span class="keyword">enum</span> <span class="keyword">class</span> <span class="identifier">CP</span> <span class="special">{</span> <span class="identifier">call</span><span class="special">,</span> <span class="identifier">put</span> <span class="special">};</span>

<span class="comment">// Assume zero annual dividend yield (q=0).</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">Price</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Sigma</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Tau</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Rate</span><span class="special">&gt;</span>
<span class="identifier">promote</span><span class="special">&lt;</span><span class="identifier">Price</span><span class="special">,</span> <span class="identifier">Sigma</span><span class="special">,</span> <span class="identifier">Tau</span><span class="special">,</span> <span class="identifier">Rate</span><span class="special">&gt;</span> <span class="identifier">black_scholes_option_price</span><span class="special">(</span><span class="identifier">CP</span> <span class="identifier">cp</span><span class="special">,</span>
                                                            <span class="keyword">double</span> <span class="identifier">K</span><span class="special">,</span>
                                                            <span class="identifier">Price</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">S</span><span class="special">,</span>
                                                            <span class="identifier">Sigma</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">sigma</span><span class="special">,</span>
                                                            <span class="identifier">Tau</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">tau</span><span class="special">,</span>
                                                            <span class="identifier">Rate</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">r</span><span class="special">)</span> <span class="special">{</span>
  <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
  <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">d1</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">log</span><span class="special">(</span><span class="identifier">S</span> <span class="special">/</span> <span class="identifier">K</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">r</span> <span class="special">+</span> <span class="identifier">sigma</span> <span class="special">*</span> <span class="identifier">sigma</span> <span class="special">/</span> <span class="number">2</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">tau</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">sigma</span> <span class="special">*</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">tau</span><span class="special">));</span>
  <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">d2</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">log</span><span class="special">(</span><span class="identifier">S</span> <span class="special">/</span> <span class="identifier">K</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">r</span> <span class="special">-</span> <span class="identifier">sigma</span> <span class="special">*</span> <span class="identifier">sigma</span> <span class="special">/</span> <span class="number">2</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">tau</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">sigma</span> <span class="special">*</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">tau</span><span class="special">));</span>
  <span class="keyword">switch</span> <span class="special">(</span><span class="identifier">cp</span><span class="special">)</span> <span class="special">{</span>
    <span class="keyword">case</span> <span class="identifier">CP</span><span class="special">::</span><span class="identifier">call</span><span class="special">:</span>
      <span class="keyword">return</span> <span class="identifier">S</span> <span class="special">*</span> <span class="identifier">Phi</span><span class="special">(</span><span class="identifier">d1</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">exp</span><span class="special">(-</span><span class="identifier">r</span> <span class="special">*</span> <span class="identifier">tau</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">K</span> <span class="special">*</span> <span class="identifier">Phi</span><span class="special">(</span><span class="identifier">d2</span><span class="special">);</span>
    <span class="keyword">case</span> <span class="identifier">CP</span><span class="special">::</span><span class="identifier">put</span><span class="special">:</span>
      <span class="keyword">return</span> <span class="identifier">exp</span><span class="special">(-</span><span class="identifier">r</span> <span class="special">*</span> <span class="identifier">tau</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">K</span> <span class="special">*</span> <span class="identifier">Phi</span><span class="special">(-</span><span class="identifier">d2</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">S</span> <span class="special">*</span> <span class="identifier">Phi</span><span class="special">(-</span><span class="identifier">d1</span><span class="special">);</span>
  <span class="special">}</span>
<span class="special">}</span>

<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> <span class="special">{</span>
  <span class="keyword">double</span> <span class="keyword">const</span> <span class="identifier">K</span> <span class="special">=</span> <span class="number">100.0</span><span class="special">;</span>                    <span class="comment">// Strike price.</span>
  <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">S</span> <span class="special">=</span> <span class="identifier">make_fvar</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">&gt;(</span><span class="number">105</span><span class="special">);</span>  <span class="comment">// Stock price.</span>
  <span class="keyword">double</span> <span class="keyword">const</span> <span class="identifier">sigma</span> <span class="special">=</span> <span class="number">5</span><span class="special">;</span>                    <span class="comment">// Volatility.</span>
  <span class="keyword">double</span> <span class="keyword">const</span> <span class="identifier">tau</span> <span class="special">=</span> <span class="number">30.0</span> <span class="special">/</span> <span class="number">365</span><span class="special">;</span>             <span class="comment">// Time to expiration in years. (30 days).</span>
  <span class="keyword">double</span> <span class="keyword">const</span> <span class="identifier">r</span> <span class="special">=</span> <span class="number">1.25</span> <span class="special">/</span> <span class="number">100</span><span class="special">;</span>               <span class="comment">// Interest rate.</span>
  <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">call_price</span> <span class="special">=</span> <span class="identifier">black_scholes_option_price</span><span class="special">(</span><span class="identifier">CP</span><span class="special">::</span><span class="identifier">call</span><span class="special">,</span> <span class="identifier">K</span><span class="special">,</span> <span class="identifier">S</span><span class="special">,</span> <span class="identifier">sigma</span><span class="special">,</span> <span class="identifier">tau</span><span class="special">,</span> <span class="identifier">r</span><span class="special">);</span>
  <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">put_price</span> <span class="special">=</span> <span class="identifier">black_scholes_option_price</span><span class="special">(</span><span class="identifier">CP</span><span class="special">::</span><span class="identifier">put</span><span class="special">,</span> <span class="identifier">K</span><span class="special">,</span> <span class="identifier">S</span><span class="special">,</span> <span class="identifier">sigma</span><span class="special">,</span> <span class="identifier">tau</span><span class="special">,</span> <span class="identifier">r</span><span class="special">);</span>

  <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"black-scholes call price = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">call_price</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span>
            <span class="special">&lt;&lt;</span> <span class="string">"black-scholes put  price = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">put_price</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span>
            <span class="special">&lt;&lt;</span> <span class="string">"call delta = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">call_price</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span>
            <span class="special">&lt;&lt;</span> <span class="string">"put  delta = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">put_price</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span>
            <span class="special">&lt;&lt;</span> <span class="string">"call gamma = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">call_price</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span>
            <span class="special">&lt;&lt;</span> <span class="string">"put  gamma = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">put_price</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span><span class="special">;</span>
  <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
<span class="special">}</span>
<span class="comment">/*
Output:
black-scholes call price = 56.5136
black-scholes put  price = 51.4109
call delta = 0.773818
put  delta = -0.226182
call gamma = 0.00199852
put  gamma = 0.00199852
*/</span>
</pre>
<h2>
<a name="math_toolkit.autodiff.h9"></a>
      <span class="phrase"><a name="math_toolkit.autodiff.advantages_of_automatic_differen"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.advantages_of_automatic_differen">Advantages
      of Automatic Differentiation</a>
    </h2>
<p>
      The above examples illustrate some of the advantages of using autodiff:
    </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
          Elimination of code redundancy. The existence of <span class="emphasis"><em>N</em></span>
          separate functions to calculate derivatives is a form of code redundancy,
          with all the liabilities that come with it:
          <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: circle; ">
<li class="listitem">
                Changes to one function require <span class="emphasis"><em>N</em></span> additional
                changes to other functions. In the 3rd example above, consider how
                much larger and inter-dependent the above code base would be if a
                separate function were written for <a href="https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks" target="_top">each
                Greek</a> value.
              </li>
<li class="listitem">
                Dependencies upon a derivative function for a different purpose will
                break when changes are made to the original function. What doesn't
                need to exist cannot break.
              </li>
<li class="listitem">
                Code bloat, reducing conceptual integrity. Control over the evolution
                of code is easier/safer when the code base is smaller and able to
                be intuitively grasped.
              </li>
</ul></div>
        </li>
<li class="listitem">
          Accuracy of derivatives over finite difference methods. Single-iteration
          finite difference methods always include a <span class="emphasis"><em>Δx</em></span>
          free variable that must be carefully chosen for each application. If <span class="emphasis"><em>Δx</em></span>
          is too small, then numerical errors become large. If <span class="emphasis"><em>Δx</em></span>
          is too large, then mathematical errors become large. With autodiff, there
          are no free variables to set and the accuracy of the answer is generally
          superior to finite difference methods even with the best choice of <span class="emphasis"><em>Δx</em></span>.
        </li>
</ul></div>
<h2>
<a name="math_toolkit.autodiff.h10"></a>
      <span class="phrase"><a name="math_toolkit.autodiff.manual"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.manual">Manual</a>
    </h2>
<p>
      Additional details are in the <a href="../../differentiation/autodiff.pdf" target="_top">autodiff
      manual</a>.
    </p>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
</div>
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